--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / Matita is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+include "datatypes/constructors.ma".
+include "nat/minimization.ma".
+include "nat/relevant_equations.ma".
+include "nat/primes.ma".
+include "nat/iteration2.ma".
+include "nat/div_and_mod_diseq.ma".
+
+definition log \def \lambda p,n.
+max n (\lambda x.leb (exp p x) n).
+
+theorem le_exp_log: \forall p,n. O < n \to
+exp p (log p n) \le n.
+intros.
+apply leb_true_to_le.
+unfold log.
+apply (f_max_true (\lambda x.leb (exp p x) n)).
+apply (ex_intro ? ? O).
+split
+ [apply le_O_n
+ |apply le_to_leb_true.simplify.assumption
+ ]
+qed.
+
+theorem log_SO: \forall n. S O < n \to log n (S O) = O.
+intros.
+apply sym_eq.apply le_n_O_to_eq.
+apply (le_exp_to_le n)
+ [assumption
+ |simplify in ⊢ (? ? %).
+ apply le_exp_log.
+ apply le_n
+ ]
+qed.
+
+theorem lt_to_log_O: \forall n,m. O < m \to m < n \to log n m = O.
+intros.
+apply sym_eq.apply le_n_O_to_eq.
+apply le_S_S_to_le.
+apply (lt_exp_to_lt n)
+ [apply (le_to_lt_to_lt ? m);assumption
+ |simplify in ⊢ (? ? %).
+ rewrite < times_n_SO.
+ apply (le_to_lt_to_lt ? m)
+ [apply le_exp_log.assumption
+ |assumption
+ ]
+ ]
+qed.
+
+theorem lt_log_n_n: \forall p, n. S O < p \to O < n \to log p n < n.
+intros.
+cut (log p n \le n)
+ [elim (le_to_or_lt_eq ? ? Hcut)
+ [assumption
+ |absurd (exp p n \le n)
+ [rewrite < H2 in ⊢ (? (? ? %) ?).
+ apply le_exp_log.
+ assumption
+ |apply lt_to_not_le.
+ apply lt_m_exp_nm.
+ assumption
+ ]
+ ]
+ |unfold log.apply le_max_n
+ ]
+qed.
+
+theorem lt_O_log: \forall p,n. O < n \to p \le n \to O < log p n.
+intros.
+unfold log.
+apply not_lt_to_le.
+intro.
+apply (leb_false_to_not_le ? ? ? H1).
+rewrite > (exp_n_SO p).
+apply (lt_max_to_false ? ? ? H2).
+assumption.
+qed.
+
+theorem le_log_n_n: \forall p,n. S O < p \to log p n \le n.
+intros.
+cases n
+ [apply le_n
+ |apply lt_to_le.
+ apply lt_log_n_n
+ [assumption|apply lt_O_S]
+ ]
+qed.
+
+theorem lt_exp_log: \forall p,n. S O < p \to n < exp p (S (log p n)).
+intros.cases n
+ [simplify.rewrite < times_n_SO.apply lt_to_le.assumption
+ |apply not_le_to_lt.
+ apply leb_false_to_not_le.
+ apply (lt_max_to_false ? (S n1) (S (log p (S n1))))
+ [apply le_S_S.apply le_n
+ |apply lt_log_n_n
+ [assumption|apply lt_O_S]
+ ]
+ ]
+qed.
+
+theorem log_times1: \forall p,n,m. S O < p \to O < n \to O < m \to
+log p (n*m) \le S(log p n+log p m).
+intros.
+unfold in ⊢ (? (% ? ?) ?).
+apply f_false_to_le_max
+ [apply (ex_intro ? ? O).
+ split
+ [apply le_O_n
+ |apply le_to_leb_true.
+ simplify.
+ rewrite > times_n_SO.
+ apply le_times;assumption
+ ]
+ |intros.
+ apply lt_to_leb_false.
+ apply (lt_to_le_to_lt ? ((exp p (S(log p n)))*(exp p (S(log p m)))))
+ [apply lt_times;apply lt_exp_log;assumption
+ |rewrite < exp_plus_times.
+ apply le_exp
+ [apply lt_to_le.assumption
+ |simplify.
+ rewrite < plus_n_Sm.
+ assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem log_times: \forall p,n,m.S O < p \to log p (n*m) \le S(log p n+log p m).
+intros.
+cases n
+ [apply le_O_n
+ |cases m
+ [rewrite < times_n_O.
+ apply le_O_n
+ |apply log_times1
+ [assumption
+ |apply lt_O_S
+ |apply lt_O_S
+ ]
+ ]
+ ]
+qed.
+
+theorem log_times_l: \forall p,n,m.O < n \to O < m \to S O < p \to
+log p n+log p m \le log p (n*m) .
+intros.
+unfold log in ⊢ (? ? (% ? ?)).
+apply f_m_to_le_max
+ [elim H
+ [rewrite > log_SO
+ [simplify.
+ rewrite < plus_n_O.
+ apply le_log_n_n.
+ assumption
+ |assumption
+ ]
+ |elim H1
+ [rewrite > log_SO
+ [rewrite < plus_n_O.
+ rewrite < times_n_SO.
+ apply le_log_n_n.
+ assumption
+ |assumption
+ ]
+ |apply (trans_le ? (S n1 + S n2))
+ [apply le_plus;apply le_log_n_n;assumption
+ |simplify.
+ apply le_S_S.
+ rewrite < plus_n_Sm.
+ change in ⊢ (? % ?) with ((S n1)+n2).
+ rewrite > sym_plus.
+ apply le_plus_r.
+ change with (n1 < n1*S n2).
+ rewrite > times_n_SO in ⊢ (? % ?).
+ apply lt_times_r1
+ [assumption
+ |apply le_S_S.assumption
+ ]
+ ]
+ ]
+ ]
+ |apply le_to_leb_true.
+ rewrite > exp_plus_times.
+ apply le_times;apply le_exp_log;assumption
+ ]
+qed.
+
+theorem log_exp: \forall p,n,m.S O < p \to O < m \to
+log p ((exp p n)*m)=n+log p m.
+intros.
+unfold log in ⊢ (? ? (% ? ?) ?).
+apply max_spec_to_max.
+unfold max_spec.
+split
+ [split
+ [elim n
+ [simplify.
+ rewrite < plus_n_O.
+ apply le_log_n_n.
+ assumption
+ |simplify.
+ rewrite > assoc_times.
+ apply (trans_le ? ((S(S O))*(p\sup n1*m)))
+ [apply le_S_times_SSO
+ [rewrite > (times_n_O O) in ⊢ (? % ?).
+ apply lt_times
+ [apply lt_O_exp.
+ apply lt_to_le.
+ assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ |apply le_times
+ [assumption
+ |apply le_n
+ ]
+ ]
+ ]
+ |simplify.
+ apply le_to_leb_true.
+ rewrite > exp_plus_times.
+ apply le_times_r.
+ apply le_exp_log.
+ assumption
+ ]
+ |intros.
+ simplify.
+ apply lt_to_leb_false.
+ apply (lt_to_le_to_lt ? ((exp p n)*(exp p (S(log p m)))))
+ [apply lt_times_r1
+ [apply lt_O_exp.apply lt_to_le.assumption
+ |apply lt_exp_log.assumption
+ ]
+ |rewrite < exp_plus_times.
+ apply le_exp
+ [apply lt_to_le.assumption
+ |rewrite < plus_n_Sm.
+ assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem eq_log_exp: \forall p,n.S O < p \to
+log p (exp p n)=n.
+intros.
+rewrite > times_n_SO in ⊢ (? ? (? ? %) ?).
+rewrite > log_exp
+ [rewrite > log_SO
+ [rewrite < plus_n_O.reflexivity
+ |assumption
+ ]
+ |assumption
+ |apply le_n
+ ]
+qed.
+
+theorem log_exp1: \forall p,n,m.S O < p \to
+log p (exp n m) \le m*S(log p n).
+intros.elim m
+ [simplify in ⊢ (? (? ? %) ?).
+ rewrite > log_SO
+ [apply le_O_n
+ |assumption
+ ]
+ |simplify.
+ apply (trans_le ? (S (log p n+log p (n\sup n1))))
+ [apply log_times.assumption
+ |apply le_S_S.
+ apply le_plus_r.
+ assumption
+ ]
+ ]
+qed.
+
+theorem log_exp2: \forall p,n,m.S O < p \to O < n \to
+m*(log p n) \le log p (exp n m).
+intros.
+apply le_S_S_to_le.
+apply (lt_exp_to_lt p)
+ [assumption
+ |rewrite > sym_times.
+ rewrite < exp_exp_times.
+ apply (le_to_lt_to_lt ? (exp n m))
+ [elim m
+ [simplify.apply le_n
+ |simplify.apply le_times
+ [apply le_exp_log.
+ assumption
+ |assumption
+ ]
+ ]
+ |apply lt_exp_log.
+ assumption
+ ]
+ ]
+qed.
+
+lemma le_log_plus: \forall p,n.S O < p \to log p n \leq log p (S n).
+intros;apply (bool_elim ? (leb (p*(exp p n)) (S n)))
+ [simplify;intro;rewrite > H1;simplify;apply (trans_le ? n)
+ [apply le_log_n_n;assumption
+ |apply le_n_Sn]
+ |intro;unfold log;simplify;rewrite > H1;simplify;apply le_max_f_max_g;
+ intros;apply le_to_leb_true;constructor 2;apply leb_true_to_le;assumption]
+qed.
+
+theorem le_log: \forall p,n,m. S O < p \to n \le m \to
+log p n \le log p m.
+intros.elim H1
+ [constructor 1
+ |apply (trans_le ? ? ? H3);apply le_log_plus;assumption]
+qed.
+
+theorem log_div: \forall p,n,m. S O < p \to O < m \to m \le n \to
+log p (n/m) \le log p n -log p m.
+intros.
+apply le_plus_to_minus_r.
+apply (trans_le ? (log p ((n/m)*m)))
+ [apply log_times_l
+ [apply le_times_to_le_div
+ [assumption
+ |rewrite < times_n_SO.
+ assumption
+ ]
+ |assumption
+ |assumption
+ ]
+ |apply le_log
+ [assumption
+ |rewrite > (div_mod n m) in ⊢ (? ? %)
+ [apply le_plus_n_r
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem log_n_n: \forall n. S O < n \to log n n = S O.
+intros.
+rewrite > exp_n_SO in ⊢ (? ? (? ? %) ?).
+rewrite > times_n_SO in ⊢ (? ? (? ? %) ?).
+rewrite > log_exp
+ [rewrite > log_SO
+ [reflexivity
+ |assumption
+ ]
+ |assumption
+ |apply le_n
+ ]
+qed.
+
+theorem log_i_SSOn: \forall n,i. S O < n \to n < i \to i \le ((S(S O))*n) \to
+log i ((S(S O))*n) = S O.
+intros.
+apply antisymmetric_le
+ [apply not_lt_to_le.intro.
+ apply (lt_to_not_le ((S(S O)) * n) (exp i (S(S O))))
+ [rewrite > exp_SSO.
+ apply lt_times
+ [apply (le_to_lt_to_lt ? n);assumption
+ |assumption
+ ]
+ |apply (trans_le ? (exp i (log i ((S(S O))*n))))
+ [apply le_exp
+ [apply (ltn_to_ltO ? ? H1)
+ |assumption
+ ]
+ |apply le_exp_log.
+ rewrite > (times_n_O O) in ⊢ (? % ?).
+ apply lt_times
+ [apply lt_O_S
+ |apply lt_to_le.assumption
+ ]
+ ]
+ ]
+ |apply (trans_le ? (log i i))
+ [rewrite < (log_n_n i) in ⊢ (? % ?)
+ [apply le_log
+ [apply (trans_lt ? n);assumption
+ |apply le_n
+ ]
+ |apply (trans_lt ? n);assumption
+ ]
+ |apply le_log
+ [apply (trans_lt ? n);assumption
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem exp_n_O: \forall n. O < n \to exp O n = O.
+intros.apply (lt_O_n_elim ? H).intros.
+simplify.reflexivity.
+qed.
+
+(*
+theorem tech1: \forall n,i.O < n \to
+(exp (S n) (S(S i)))/(exp n (S i)) \le ((exp n i) + (exp (S n) (S i)))/(exp n i).
+intros.
+simplify in ⊢ (? (? ? %) ?).
+rewrite < eq_div_div_div_times
+ [apply monotonic_div
+ [apply lt_O_exp.assumption
+ |apply le_S_S_to_le.
+ apply lt_times_to_lt_div.
+ change in ⊢ (? % ?) with ((exp (S n) (S i)) + n*(exp (S n) (S i))).
+
+
+ |apply (trans_le ? ((n)\sup(i)*(S n)\sup(S i)/(n)\sup(S i)))
+ [apply le_times_div_div_times.
+ apply lt_O_exp.assumption
+ |apply le_times_to_le_div2
+ [apply lt_O_exp.assumption
+ |simplify.
+
+theorem tech1: \forall a,b,n,m.O < m \to
+n/m \le b \to (a*n)/m \le a*b.
+intros.
+apply le_times_to_le_div2
+ [assumption
+ |
+
+theorem tech2: \forall n,m. O < n \to
+(exp (S n) m) / (exp n m) \le (n + m)/n.
+intros.
+elim m
+ [rewrite < plus_n_O.simplify.
+ rewrite > div_n_n.apply le_n
+ |apply le_times_to_le_div
+ [assumption
+ |apply (trans_le ? (n*(S n)\sup(S n1)/(n)\sup(S n1)))
+ [apply le_times_div_div_times.
+ apply lt_O_exp
+ |simplify in ⊢ (? (? ? %) ?).
+ rewrite > sym_times in ⊢ (? (? ? %) ?).
+ rewrite < eq_div_div_div_times
+ [apply le_times_to_le_div2
+ [assumption
+ |
+
+
+theorem le_log_sigma_p:\forall n,m,p. O < m \to S O < p \to
+log p (exp n m) \le sigma_p n (\lambda i.true) (\lambda i. (m / i)).
+intros.
+elim n
+ [rewrite > exp_n_O
+ [simplify.apply le_n
+ |assumption
+ ]
+ |rewrite > true_to_sigma_p_Sn
+ [apply (trans_le ? (m/n1+(log p (exp n1 m))))
+ [
+*)
\ No newline at end of file
projects / helm.git / blobdiff